Speed of coming down from infinity for $\Lambda$-Fleming-Viot initial support

The $\Lambda$-Fleming-Viot process is a probability measure-valued process that is dual to a $\Lambda$-coalescent that allows multiple collisions. In this paper, we consider a class of $\Lambda$-Fleming-Viot processes with Brownian spatial motion and with associated $\Lambda$-coalescents that come down from infinity. Notably, these processes have the compact support property: the support of the process becomes finite as soon as $t>0$, even though the initial measure has unbounded support. We obtain asymptotic results characterizing the rates at which the initial supports become finite. The rates of coming down are expressed in terms of the asymptotic inverse function of the tail distribution of the initial measure and the speed function of coming down from infinity for the corresponding $\Lambda$-coalescent.